Editing Natural transformation (section)

===Determinant===
{{See also|Determinant#Square matrices over commutative rings}}

Given [[Commutative ring|commutative rings]] <math>R</math> and <math>S</math> with a [[ring homomorphism]] <math>f : R \to S</math>, the respective groups of [[Invertible matrix|invertible]] <math>n \times n</math> matrices <math>\text{GL}_n(R)</math> and <math>\text{GL}_n(S)</math> inherit a homomorphism which we denote by <math>\text{GL}_n(f)</math>, obtained by applying <math>f</math> 
to each matrix entry. Similarly, <math>f</math> restricts to a group homomorphism <math>f^* : R^* \to S^*</math>, where <math>R^*</math> denotes the [[group of units]] of <math>R</math>. In fact, <math>\text{GL}_n</math> and <math>*</math> are functors from the category of commutative rings <math>\textbf{CRing}</math> to <math>\textbf{Grp}</math>. 
The [[determinant]] on the group <math>\text{GL}_n(R)</math>, denoted by <math>\text{det}_R</math>, is a group homomorphism 

: <math>\mbox{det}_R \colon \mbox{GL}_n(R) \to R^*</math>
which is natural in <math>R</math>: because the determinant is defined by the same formula for every ring, <math>f^* \circ \text{det}_R = \text{det}_S\circ \text{GL}_n(f)</math> holds. This makes the determinant a natural transformation from <math>\text{GL}_n</math> to <math>*</math>.